The 18-year-old's contract expires next year and his decision to reject a new deal was made on Thursday after talks between the player, his agent Mino Raiola and the club.
"Raiola informed us that Donnarumma has made a definitive decision not to renew the contract with Milan," Fassone said in a statement.
"We were taken by surprise, as we had hoped Donnarumma would be the goalkeeper taking Milan into the future. Naturally, the decision leaves a bitter taste in the mouth, but now we have to go forward."
The Italy international made his under-21 debut in 2015 and was called up to the senior side in 2016.
(Reporting by Aditi Prakash in Bengaluru; Editing by Peter Rutherford)
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